LECTURE 5 1. Isotropic Grassmannians In this section, we discuss the geometry of isotropic Grassmannians. Let V be an n-dimensional complex vector space. Let Q be a non-degenerate quadratic form. The form Q may either be symmetric or skew-symmetric. Since the rank of a skew- symmetric form is always even, in the latter case, we must assume that n is even. There are some differences in the discussion depending on whether Q is symmetric or skew-symmetric. Furthermore, when Q is symmetric, there are slight variations in detail depending on whether n is even or odd. We will now discuss these cases separately. 1.1. Preliminaries on quadrics. Let Q be a smooth quadric hypersurface in n−1 n P . Set m = b 2 c. The largest dimensional linear spaces contained in Q have projective dimension m−1. If n is odd, then the maximal dimensional linear spaces m(m+1) on Q form an irreducible family of dimension 2 : If n is even, then the maximal dimensional linear spaces contained in Q form two isomorphic families of dimension m(m−1) 2 : Two linear spaces belong to the same irreducible component if and only if their dimension of intersection is equal to m − 1 modulo 2 (see [GH] p. 735). More generally, we will be interested in linear spaces on quadric hypersurfaces with singularities. A quadric hypersurface in Pn−1 of corank r (or, equivalently, with a singular locus of dimension r − 1) is the cone over a smooth quadric hy- persurface in Pn−1−r with vertex an (r − 1)-dimensional projective linear space. If Q is a quadric hypersurface of corank r in Pn−1, then the largest dimensional n−r−2 linear space on Q has dimension b 2 c+r: The space of linear spaces of maximal dimension on Q is irreducible if n − r is odd and has two irreducible components if n−r−3 n−r−2 n−r is even. Setting l = 2 in the former case and l = 2 in the latter case, the dimension of each irreducible component of the space of maximal dimensional (l+1)(l+2) l(l+1) linear spaces is 2 and 2 , respectively. In the latter case, two linear spaces belong to the same irreducible component if and only if their dimension of intersection is equal to l + r modulo 2. These claims follow from the previous paragraph since Q is a cone over a smooth quadric hypersurface in Pn−1−r. Notation 1.1. Denote the Fano variety of s-dimensional projective linear spaces n−1 r contained in a quadric hypersurface Q 2 P of corank r by Fs;n(Q). Let Q ⊂ Pn−1 be a quadric hypersurface of corank r. Let s be a positive integer n−r−2 less than or equal to b 2 c + r. Consider the incidence correspondence of pairs of a point p of Q and an s-dimensional linear space containing p: r I = f (x; [Λ]) j x 2 Λ ⊂ Q g ⊂ Q × Fs;n(Q): The automorphism group of Q acts transitively on the smooth points of Q. The s-planes that contain a smooth point p lie in the tangent linear space H at p. Q\H is a quadric hypersurface of corank r +1. The intersection with a hyperplane 1 complementary to p is a quadric hypersurface of corank r and intersects all the s-planes containing p in an (s − 1)-dimensional linear space. We conclude that the space of s-dimensional linear spaces containing p has the same dimension as the space of (s−1)-dimensional linear spaces lying on a quadric hypersurface in Pn−3 of corank r. Therefore, by induction, we can calculate the general fiber dimension of the projection of I to Q and determine the dimension of I. The second projection r maps I onto Fs;n(Q) with fiber dimension s. We thus obtain a recursion relation r for the dimension of Fs;n(Q). A priori we need to check that the s-dimensional linear spaces that intersect the n−r−2 vertex in dimension greater than s − 1 − b 2 c do not form another irreducible r component (potentially of different dimension) of Fs;n(Q). It is easy to see that linear spaces that intersect the vertex in larger than the expected dimension are limits of linear spaces that intersect the vertex in the expected dimension. Observe that every linear space on a quadric is contained in a maximal dimensional linear space. Take a linear space Λ that intersects the vertex in the linear space Ω. Assume that the dimension of Ω is larger than expected. Take a linear space ∆ in n−r−2 Λ complementary to Ω. Take a linear space Γ of dimension b 2 c which contains ∆, but does not intersect the vertex of Q. Since the Grassmannian of s-planes in the span of Γ and Ω is irreducible, the claim follows. n−r−2 In case s < 2 , the space of s-dimensional linear spaces on Q is irreducible. n−r−2 r+1 If s ≥ 2 the recursion stops when we obtain a quadric of rank r in P or Pr with multiplicity 2. The former case occurs if n − r is even and the latter case occurs if n − r is odd. This allows us to calculate the dimensions of the spaces of n−r−2 s-dimensional linear spaces on Q recursively. It also proves that when s ≥ 2 , the spaces of s-dimensional linear spaces on Q is irreducible if n − r is odd and has two components if n − r is even. We have thus proved the following: n−1 n−r−2 Lemma 1.2. Let Q be a quadric hypersurface in P of corank r. If s < 2 , r then Fs;n(Q) is irreducible of dimension 2n − 3s − 4 (s + 1) : 2 n−r−2 r If s ≥ 2 and n − r is even, then Fs;n(Q) has two irreducible components each of dimension n − 2s + r − 2 (n − r − 2) (n − r) (s + 1) + : 2 8 n−r−2 r If s ≥ 2 and n − r is odd, then Fs;n(Q) is irreducible of dimension n − 2s + r − 3 (n − r − 1) (n − r + 1) (s + 1) + : 2 8 1.2. Preliminaries on orthogonal Grassmannians. Let W be an n-dimensional vector space endowed with a non-degenerate, symmetric, bilinear form Q. Set n m = b 2 c. Let 0 < k ≤ m denote a positive integer. Let OG(k; n) denote the k- dimensional subspaces of W isotropic with respect to the form Q, unless n = 2k. In the latter case, the parameter space of k-dimensional isotropic subspaces of W has two isomorphic irreducible components. OG(k; n) denotes one of these irreducible components. The orthogonal Grassmannian OG(k; n) is isomorphic to one irreducible compo- 0 nent of the Fano variety Fk−1;n(Q) of (k − 1)-dimensional projective linear spaces 2 on a smooth quadric hypersurface. The non-degenerate quadratic form Q defines the smooth quadric hypersurface in Pn−1. A linear space is isotropic with respect to Q if and only if its projectivization is contained in the quadric hypersurface defined by Q. In particular, by the discussion in x1.1, the dimension of OG(k; n) is k(2n − 3k − 1) 2 The cohomology of OG(k; n) is generated by the classes of Schubert varieties. There are minor differences in the cohomology of OG(k; n) depending on the parity of n due to the fact that when n is even, the half-dimensional isotropic subspaces form two connected components. For even n, the notation has to distinguish be- tween these two connected components. For simplicity, we will first discuss the case of odd n, then describe the necessary modifications for even n. We begin by describing the Schubert varieties in OG(k; 2m + 1). Let λ denote a sequence m ≥ λ1 > λ2 > ··· > λs > 0 of strictly decreasing integers, where s ≤ k. Given λ, there is an associated sequence ~ ~ m − 1 ≥ λs+1 > ··· > λm ≥ 0 of strictly decreasing integers defined by requiring that there does not exist any ~ parts λi for which λj +λi = m. In other words, the associated partition is obtained by removing the integers m − λ1; : : : ; m − λs from the sequence m − 1; m − 2;:::; 0. For example, if m = 6, then the partition associated to (6; 4) is (5; 4; 3; 1). The Schubert varieties in OG(k; 2m + 1) are parameterized by pairs (λ, µ), where λ is a strictly decreasing partition of length s and µ m − 1 ≥ µs+1 > µs+2 > ··· > µk ≥ 0 is a subpartition of λ~ (i.e., the parts of µ are a subset of the parts of λ~) of length k − s. We will call such pairs of partitions allowed pairs. Observe that for maximal isotropic Grassmannians OG(m; 2m + 1), the partition µ = λ~ is uniquely deter- mined by the partition λ. Consequently, in the literature it is standard to omit the sequence µ and parametrize Schubert varieties by strict partitions λ. We will find it useful to record the dimensions of all the flag elements where a jump in dimen- sion occurs, so we add µ to the notation. For non-maximal Grassmannians there are several notations in use. The advantage of our notation is that it minimizes the amount of calculation needed to determine the dimensions of the flag elements where a jump in dimension occurs.